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V .n". .100 .V: 0 c ... . . 40 . . . . . . . . V00. . 0 . V . 1 l 40 V 0 ..0 — 0 I I - 0|. 0 I I'. I r W ! ' 2 LIBRARY 110001 Michigan State University This is to certify that the dissertation entitled A MICROSCOPIC HYPER-SPHERICAL MODEL OF TWO-NEUTRON HALO NUCLEI presented by Ivan Brida has been accepted towards fulfillment of the requirements for the Ph.D. degree in Physics and Astronomy (rrwajar Professor" s~§g\nature EJ434011 Z 00% Date MSU is an Affirmative Action/Equal Opportunity Employer PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 K:/Prolecc&Pres/ClRCIDateDue indd A MICROSCOPIC HYPER-SPHERICAL MODEL OF TWO-NEUTRON HALO NUCLEI By Ivan Brida A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Physics and Astronomy 2009 A Mimi \I-I' llin'i' ll morpwrutm I E‘ijilaifih iii it Gaillfifilitln. I‘ am mm in nu hi'pt‘ir-rmiiiil ;: are? usml fur tl In the“ pm 6H6. Tlli‘ (rm Eil'liliiii'ili is ”N mule-ls and in 1Hi}. railll itliil dwriptiun HI. Ffliiiluwl in t Crucial ml h illxll arr: liwn ext: I'I‘iHii‘Iliri'in II ABSTRACT A MICROSCOPIC HYPER-SPHERICAL MODEL OF TWO-NEUTRON HALO ' NUCLEI By Ivan Brida We have developed a microscopic cluster model of light two neutron halo nuclei that incorporates the few-body asymptotics in full extent. The wavefunction of the system consists of a core and two valence neutrons. The core is given in terms of correlated Gaussians. The three-body dynamics between the core and valence neutrons is taken into account by means of the hyper-spherical functions containing an exponentially decaying hyper-radial part. To avoid the spurious motion of the center of mass, Jacobi coordinates are used for the entire system. In the present work, the model is applied to the lightest two-neutron halo nucleus, 6He. The central Minnesota nucleon-nucleon interaction with and without a spin-orbit addition is used to bind the nucleus. The results are compared to those obtained in other models and to experimental data. Basic structural observables, such as binding relative to 4He, radii and one-body densities are in agreement with other models. The microscopic description of the core allows us to test the efficiency of Pauli projection techniques employed in the few-body models. We demonstrate that proper antisymmetrization is crucial to bind 6He against three-body break-up. Overlap functions between 6He and 4He have been extracted with the aim of reaction calculations involving 6He. In particular, two-neutron transfer reaction p(6He, 4He)t at 25MeV/ A is studied. I intuit with mt“ ()II'II llil'ti“ ' $97311th ‘Ilil “ll I: nae-Int-t-rs l Grrg'T‘ ll. I \tuiiilil annual .\ln I’afii‘ifinimi Xi'xi. l Stair lilill'i‘ mt‘lll ii! iii: Whirl fail a the Dignim lam grei iii-r» Hit i, _: Pitts rm im' riililli’x'w] it: and Angel”? dim. llll'liti MIMI). and LN lJlli ml ba'lJTlii‘rg ACKNOWLEDGMENTS I would first like to acknowledge my advisor Filomena Nunes. I admire her patience with me—a stubborn student of hers, the freedom she gave me to develop and work on my own ideas even when she did not find them useful, and her guidance and jokes throughout the years. Being her first graduate student, I hope her next students will be less stubborn and will give her less headaches. I would also like to thank my guidance committee members Edward Brown, Sekhar Chivukula, Remco Zegers, Vladimir Zelevinsky, and Gregers Hansen for their comments and suggestions. I would like to thank Robert Wiringa and Steve Pieper for discussions on the vari- ational Monte Carlo method. I thank Kalman Varga for discussions on the stochastic variational model and for providing me his computer codes. Next, I would like to thank the Department of Physics and Astronomy at Michigan State university, the NSCL and especially the theory group for providing a great environ- ment to work in. Special thanks goes to Shari Conroy without whom the theory group would fall apart in chaos. The financial support of the National Science Foundation and the Department of Energy is also acknowledged. I am greatly indebted to the other members of my cohort: Matt Amthor, Jon Cook, Wess Hitt, Andy Rogers, and Terrance Strother. I will never forget all their inappropriate jokes on my behalf and the long nights we spent working on homeworks. Special thanks is addressed to my current and former office mates Jeremy Armstrong, Biruk Gebremariam, and Angelo Signoracci. There are a lot of thank yous for other graduate students and post- docs, including Ania Kwiatkowski, Giuseppe Lorusso, Rhiannon Meharchand, Michal Mocko, and Arnau Rios. Last, but certainly not least, I express my gratitude to my family, my parents and my brothers for everything they have done for me and for helping me to make it to this point in my life. iii :il “7“ Contel List of Till) List of Figi‘ l Introduciiu l.l lldliiili 1.2 Tli'iI-ni 1.3 OU'H'li' ii (iv-m. ' l3 Mullml 1.6 Outline- 2 Valence [)éil'l 2.1 (‘llllflllllw 3.2 Ollll'fll. 2-3 ”Lilli I. “2.3.1 l: 2.3.2 1' 2.3.3 [5. 2.3.5 (2 2'4 b“Mum. 2.1.1 1,... 2-4-2 Ill" 2.4.3 it: Contents List of Tables ................................... List of Figures .................................. 1 Introduction 1.1 Halo in nuclei ................................. 1.2 Two—neutron halo nuclei: 6He and 11Li ................... 1.3 Overview of 6He and 11Li: experiments ................... 1.4 Overview of 6He and 11Li: structure theory ................. 1.5 Motivation for present work ......................... 1.6 Outline ..................................... 2 Valence part 2.1 Coordinates and bases ............................ 2.2 Other ingredients of the three-body model ................. 2.3 11Li in the three—body model ......................... 2.3.1 Introduction .............................. 2.3.2 10Li .................................. 2.3.3 Interactions .............................. 2.3.4 Results ................................. 2.3.5 Conclusions .............................. 2.4 6He in the three-body model ......................... 2.4.1 Introduction .............................. 2.4.2 Interactions .............................. 2.4.3 Results ................................. 3 Core 3.1 Stochastic variational model ......................... 3.2 4He in the stochastic variational model ................... 3.2.1 Interactions .............................. 3.2.2 Results ................................. 4 MiCH: final assembly 4.1 Core and valence together .......................... 4.2 Variational Monte Carlo ........................... 4.2.1 Monte Carlo essentials ........................ 4.2.2 Can we trust ourselves? ....................... 4.2.3 Wavefunction optimization ...................... iv 55 58 60 60 66 74 81 2.31 - 0‘ I'v' Jun-In: ...-— 0‘ Summary 6.1 531mm. 62 ”Mint Implement ..‘11 Ltii'iili .12 (Ila-Int? Further t0: 8.1 .l‘l’lli ill 8.) {\‘IIIIIii C Comparat i‘ D Wavefum‘ti 5 6He in MiCH 5.1 Antisymmetrization effects in 6He ...................... 5.2 Converged 6He ................................ 5.3 Overlap functions ............................... 5.4 Two-neutron transfer reactions ....................... 6 Summary and outlook 6.1 Summary ................................... 6.2 Outlook .................................... A Implementation details A.1 Local representation of wavefunction .................... A.2 Operators ................................... B Further tests B.1 Triton tests and the story of bad points ................... B.2 Additional tests and checks ......................... C Comparative optimization on two independent random walks D Wavefunction normalization Bibliography 87 87 93 101 113 118 118 121 123 123 125 127 127 132 134 138 140 r 31 (\Q\ Pi" (1H! Stir] Kilt Slim PD i List of Tables 2.1 2.2 5.1 5.2 5.3 5.4 5.5 Probabilities of dominant components in the T Jacobi basis in 6He . . . 43 Probabilities of dominant components in the Y Jacobi basis in 6He . . . 44 Combinations of quantum numbers for K = 0 and K = 2 channels . . . . 89 Squares of the Raynal-Revai coefficients for K = 2 valence channels . . . 89 Energies and radii of 6He obtained in different models and from experiment 97 SpectrOSCOpic factors of the five dominant overlap channels in 6He . . . . 105 Probabilities of overlap channels in the 4He + n + n decomposition of 6He 110 vi *2 5' RM List of . 11 Lower gm 1. N1 . L2 bfiiiillt1.i v- :1 n. ‘I .. ' .11 .I‘tttftiliiiili c ‘ it'll'illllit‘ ..3 Hit-"11118 iii- .4 T1161HW'.‘ :3 Di-ptiis [at 2,6 Bimiinu v: 2..Binr1ii.gvr Dt‘pi‘nill'llt 2.9 T111T‘t‘-l)l H1 2111 R3125 riiii'li 2.11 Proliilliiiiti. 2.12 Dr-penrl t‘lu' 21'? 2’ HYDPHM; ,7 'IdI‘UbI ('(Irfl‘ ‘ dlil' d1 (li'Iii List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 3.1 3.2 3.3 4.1 4.2 4.3 4.4 Lower part of the chart of nuclei ....................... 3 Schematic representation of structure models of 6He ............ 8 Jacobi and hyper-spherical coordinates for a three-body system ..... 17 Coordinates used to define two-body potentials in the three-body model . 23 Radius of the 9Li-n interaction as a function of deformation ....... 32 The lowest energy levels in 10Li produced by the core-n interaction . . . 34 Depths of the fitted 9Li-n interaction as a function of deformation. . . . . 34 Binding energy of bound states in 10Li as a function of deformation . . . 35 Binding energy of continuum states in 10Li as a function of deformation . 35 Dependence of the three-body binding energy of 11Li on Km“ ...... 36 Three-body binding energy of 11Li as a function of deformation ..... 37 Rms matter radius of 11Li as a function of deformation .......... 37 Probabilities of structural components in 11Li as a function of deformation 38 Dependence of the three-body binding energy of 6He on Kmam ...... 42 Hyper-radial dependence of dominant channels in 6He ........... 44 Jacobi coordinates for a system of four identical particles ......... 50 Radial dependence of the central Minnesota force with u = 1.015 ..... 58 Convergence of the binding energy of the MN and MN-SO 4He ...... 59 Effects of correlations in the Metropolis algorithm on local energies . . . 76 Flow chart for the decorrelated Metropolis algorithm ........... 77 Effects of decorrelated sampling on auto—correlation coefficient ...... 77 Effects of bunching on data from Figure 4.3 ................ 79 vii al 4.5 II a" C 4.6 131 5.2 ('t 5.3 Bi] 5.1 1‘2, 5.5 D i: 5.6 113' 5.7 in] 5.9 .-\~\ 39 ('tif‘ I 5.111 I11“: 1,...“ WJ '31 1 (‘fi i: 8-1 Pitt} 82 Bilii Ilili‘lfll’fi in 4.5 Running mean energy and energy error estimate .............. 79 4.6 Block values for data from Figure 4.3 .................... 80 5.1 Three-body binding energy of valence channels as a function of po . . . . 90 5.2 Convergence of the three-body binding energy of 6He ........... 94 5.3 Binding energy and rms proton radius of 6He as a function of po ..... 95 5.4 Point proton and neutron density distributions in 6He ........... 100 5.5 Point proton and neutron density distributions in 6He in linear scale . . . 102 5.6 Hyper-radial dependence of overlap channels in 6He in linear scale . . . . 106 5.7 Hyper—radial dependence of overlap channels in 6He in logarithmic scale . 107 5.8 Asymptotical hyper-radial behavior of K = O s-waves ........... 109 5.9 Correlation density plot for the ground state of the MN—SO 6He ..... 112 5.10 Hyper-angular probability in the K = 2 s-waves overlap channel ..... 113 5.11 Cross-section of the p(6He,4He)t reaction at 25 MeV/ A .......... 116 B1 Properties of simple tritons ......................... 130 B.2 Binding energy and rms matter radius of a triton ............. 132 Images in this dissertation are presented in color. viii Chapter 1 Introduction Atomic nuclei represent self-bound ensembles of strongly interacting fermions. Experi- mental and theoretical explorations of the chart of nuclei have revealed many intriguing features of nuclear matter. Among them, a structural hallmark—the nuclear halo—has been found in the realm of light nuclei near the limits of particle stability. In general, the halo phenomenon is a threshold effect occurring in loosely bound systems, in which particles are held in short-range potential wells. In favorable circum— stances, a barely trapped particle or particles (or a cluster of particles) may tunnel out into the classically forbidden region. This “leakage” populates very dilute and fragile structures near particle emission thresholds. The more loosely the halo particles are con- fined, the more clearly “the halo stratosphere” is developed. Besides nuclear physics, halo systems are known or expected to exist in other branches of physics as well. One of the most extended halo systems known to exist is the atomic helium dimer 4H82 which is about ten times larger than a typical diatomic molecule and is bound by only about 10‘7 eV [1]. Halo states have been predicted or experimentally observed for a range of other systems, such as 3He—3He-39K [2], positron-atom complexes [3], hyper-nuclei such as 31H [4] among others. A comprehensive review of halo systems can be found for example in [5]. a ii A“ 1 1.1 H The gilt-111131 Ilse) PINK" energies it it I (‘flllSlllt‘fatl ulx furn'mtiun Hi- In a first of The swirl: nurlei ("an it. orbiting it Ii]: 1n quantum l‘ l. the prui llll‘lllllt.‘ 3. and Illi' a gin-n ; 1t has 1mm in” it Illif‘ "‘llt‘I‘: small. in 011110 m. b' 111" llétli ) (7 and 1]“, 1 t() (1(l\-( l]( l 1.1 Halo in nuclei The quantum-mechanical tunneling present in halo nuclei produces unexpected effects. The energy needed to remove halo nucleons is drastically less than particle separation energies for typical nuclei. Nuclear radii are enhanced; matter and charge radii may differ considerably. There is evidence that few-body effects may become crucial, leading to the formation of cluster structures beyond the reach of mean field theories. In a first approximation, the spatial separation of particles in the halo from the rest of the system justifies a simplified description with only a few active constituents. Halo nuclei can be thought of in terms of a few (typically one or two) single halo nucleons 1 orbiting a tightly bound core, thus implying a major role of single-particle properties. In quantitative terms, it has been assessed [6,7] that for a quantum halo to develop, 1. the probability to find halo particles in the forbidden region beyond the classical turning point should be more than 50%, 2. and the core-halo configuration should occur with more than a 50% probability in a given system. It has been argued [8,9] that for a nucleus to meet these criteria: a. the energy needed to separate the halo part from the rest of the nucleus should be small, more precisely less than about 2 MeV 14—2/3, with A being the mass number of the nucleus, b. the halo nucleons should occupy s— or p—angular momentum orbits around the core, c. and the proton number of the nucleus should not exceed ten or so for a proton halo to develop. For three-body halo states containing two loosely bound nucleons, the condition b. should be supplemented by a requirement of hyperlmomentum2 K = 0 or 1. The formation of 1Here, we do not consider less straightforward cluster divisions with tightly bound subgroups of nucleons, such as 9Be consisting of two 0: clusters glued together by a neutron. 2To be introduced in Chapter 2. ..al A figure 11: scum-s. In I. [3 explaini‘li a chargul 1m nuclei in Fl: From Ilu' neutron-rub “30 l: “'13. 13(11: 13C 4 ”iii g: 3m r". trill-3 tiff (‘XIN II da-ay 1i 'Ilqill. of ordinary. ti- In IIIIt'lt'ur mil-1‘2 WIT 11 m I .1 ' f I \ J ' Ll -\ 1.1 + “1'2," till the at 1136145 .1 thanks Ii Mitten halo I!" r,. [H . ”I (“miplun y Proton Number Neutron Number p I pln - Borromean ° l [j pln - Halo Figure 1.1: Lower part of the chart of nuclei. Stable nuclei are represented by black “ ,7 squares. In this figure, p stands for a proton, “n” for a neutron. The term ”Borromean” is explained in Section 1.2. a charged halo is hindered by the Coulomb barrier. These conditions naturally favor light nuclei in Figure 1.1 to populate halo states. From the modern perspective, the best established nuclear halos live among light neutron-rich nuclei. Examples of one-neutron halo nuclei include the ground states of 11Be (2 10Be + n) [10] and 19C (= 18C + n) [11], excited states in 12B (= 11B + n) and 13C (= 12C + n) [12] and several possible candidates, such as 31Ne (= 30Ne + n) and 40A1 (= 39A1 + n) [5]. In one-neutron halos, the tail of the relative core-n wavefunction falls off exponentially with the distance between the core and the extra neutron. The decay length, determined by the neutron separation energy, is typically 4—5 times that of ordinary, tightly bound nuclei [5]. In nuclear physics, the most obvious three-body halo candidates are light drip-line nuclei with two neutrons encircling a core. Among them, 6He (2 4He + n + n) and 11Li (= 9Li + n + n) are stereotypical prototypes of nuclear halo systems [13], and they enjoy all the attention of the present work. 11Li is considered the prima donna of all halo nuclei thanks to its very small two-neutron separation energy 378 keV [14]. Other two- neutron halo nuclei include 14Be (2 12Be + n + n) [15], possibly 22C (= 20C + n + n) [16], and other candidates [5]. For completeness, we should mention other nuclei in which some sort of halo may 3 8351' be (lt’X't‘lifl'f ]i]((l]\' m 1H deuti’rH“ 1‘ enerEV {7 states 17 belimnl tn pui_‘>p111‘dilt in i bwn sewn II states. Iii-i}; As an exam Illt’illlilll-llut.‘ “uric ('Iilit‘ll exist at lllt" i in lllt’ [in particular (it: 1.2 T“ Apart in im nuclei mm b 111 (illntl. awn t Bill'tffllli’itll is a way that If example. in.- l .- ,1 - “(511- TlllS (K 1r1~ be deveIOped. In the deuteron, for example, the proton (p) and the neutron are very likely to be found outside the range of the strong interaction. The binding energy of the deuteron (-2.2 MeV) is in absolute value small compared to a typical nucleon separation energy (7—8 MeV), arguably making the deuteron the forerunner of all nuclear halo states [17]. On the neutron-rich side of the chart of nuclei, 8He contains four neutrons believed to form a neutron skin around the 4He core [18]. On the proton-rich side, the population of halo nuclei is decimated by the Coulomb barrier. Hints of a proton halo have been seen in 8B (= 7Be + p) [19], 17'Ne (2 15O + p + p) [20], and some other nuclear states. Reference [5] contains a more complete list of possible halo states in light nuclei. As an example of theoretical studies on the existence of halo effects in heavier nuclei, medium-mass even-even nuclei have been scrutinized in [21, 22]. The authors of these works concluded that on the large scale the halo phenomenon is very rare and can only exist at the very limit of neutron stability. In the present work, however, we shall focus only on light two-neutron halo nuclei, in particular on 6He and 11Li. 1.2 Two-neutron halo nuclei: 6He and 11Li Apart from possessing all of the peculiar halo features, the known two-neutron halo nuclei including 6He and 11Li are Borromean, meaning that the system core + n + n is bound, even though the binary subsystems core + n and n + n are unbound. The term Borromean is adopted after a heraldic symbol of three rings which are joined in such a way that if any one is broken, all three become free [13]. In the helium chain, for example, 4He binds two extra neutrons, but not one, and the di-neutron is unbound as well. This odd-even staggering is merely a consequence of nucleon-nucleon correlations. One then deals meticulously with two correlated neutrons revolving around a core in the low density regime. Thus, these nuclei are ideal playgrounds to study neutron correlations in an almost proton-free environment. It is possible that these nuclei give rise to the so- called Efimov states [23,24]. p 3' AQ‘ c"_‘ 777‘ + V__-" The BU: cut? and {h [0:102:03 1“ V 1 7 into 8 C" ”‘5‘ ‘1 distant)?" “'5 Iillln- [111‘ V'Il‘ the two will These IIEIIIIII' aiwtmt l 3 I?! dramatic in 1 Ont" tili‘tt £111 astmptntir [1 DUI? in ii Single l)iitllltl In 1" ‘ ..lt‘ Of Jill' Is. Dc» studiml. I dll'dll'Inly (if [j p- ,- ,. '7 .Mitdttuiis iii ’19 L1 ‘ “I‘M rPW.‘ 81mm ] ' -' ‘ ll-‘lurli‘ L sit «2 .r ' The Borromean nature of 6He and 11Li implies that, even at large distances, the core and the valence particles are correlated with no bound binary admixtures. Asymp— totically, the wavefunction vanishes exponentially with a decay rate depending on the three-body binding energy, i.e. on the amount of energy needed to break the nucleus up into a core and two free neutrons. The inverse of the decay rate gives a typical “three-body distance” within the nucleus, which is about 7.5 fm in 11Li. For better visual apprecia- tion, this value corresponds to a di-neutron at distance about 6 fm from a 9Li core or to the two neutrons being on opposite sides of the core at mutual distance of about 11 fm. These numbers are to be compared with the range of the nucleon-nucleon interaction of about 1—2 fin and also with the 2.32 fm radius of the 9Li core [13]. The situation is less dramatic in 6He due to its larger two-neutron separation energy3 of about 970 keV [25]. One then anticipates that many properties of 6He and 11Li will depend chiefly on the asymptotic part of the wavefunction. Due to the proximity of particle emission thresholds, 6He and 11Li support only a single bound state, the ground state. Moreover, these nuclei are short-lived; the half- life of 6He is 806.7 ms [25] and that of 11Li is even shorter at about 8.8 ms [26]. To be studied, these nuclei have to be produced artificially. Most information about the anatomy of nuclear halos has been obtained in reaction processes leading to continuum excitations and ultimately to the destruction of the investigated nuclei. It is useful to put the most rewarding experimental methods into their historical context. In the following short historical overview, we focus mainly on “Li, but some of the experiments have been carried out for other halo nuclei including 6He. 1.3 Overview of 6He and 11Li: experiments The history of two-neutron halo nuclei started with the discovery of 6He back in the 19308 [27]. It took three more decades to produce 11Li for the first time [28]. Current 3In what follows, the two-neutron separation energy is taken as an absolute value of the three-body binding energy, and the two terms will be used interchangeably. fifi\a I intents! in I minus. I: with otiliIM were film“ i (iinsistinz U by a llli‘dhlll Consistvnt \' certainty pr: mmliél alw s Soon after. I high :33: and ('ildrgi"-i‘xi '11. I implying t1..~t ‘lf‘lliltf‘ tlw II that only :1. I lflti‘f'dt’tiuti (-1 Scattered Mm smaller by a], .3331.“ of tlti‘ (I] [11 paramuiq. [16051112 .341” ”2‘10 S[flu-rim inn; 9L1 am 21 251136: exrmn r1 :ppf’n‘d in ".11 l 95.30111311an in listening up a I The (ta it. .43 1 TI.) Tr .—‘ ..JJ. Tilt“; interest in nuclear halos, however, was sparked by the advent of modern radioactive beam facilities. In 1985, the interaction cross section of helium and lithium isotopes colliding with ordinary nuclear targets was measured [29,30]. The surprisingly large values for 11Li were soon interpreted as a consequence of extended neutron densities, a neutron halo, consisting of a di-neutron coupled to a 9Li core [31]. This speculation was later supported by a measurement of the momentum distribution of 9Li after the break-up of 11Li [32]. Consistent with the di—neutron model, large spatial extent of the halo was, through the un- certainty principle, reflected by narrow relative momentum distributions. The di-neutron model also suggested large two-neutron removal cross sections via Coulomb dissociation. Soon after, the cross sections of electromagnetic dissociation of 11Li on high-Z targets at high [33] and low beam energies [34] were found to reach anomalously large values. Later, charge-exchange cross sections of 829211Li were measured to be about the same [35], thus implying that the 9Li core is little disturbed in 11Li. One of the first attempts to indirectly deduce the neutron density profile of 11Li can be found in [36]. The authors concluded that only density distributions with very long tails consistently reproduce the observed interaction cross-sections. Furthermore, the angular distributions of 9Li and 11Li nuclei scattered elastically from protons are similar, but the elastic scattering cross-section is smaller by about a factor of two for 11Li [37]. In data analysis, both real and imaginary parts of the optical potentials had to be changed considerably for 11Li compared to global fit parameters, in order to account for break-up due to the extended tail of the neutron density. fl-decay represents an interesting alternative for extracting information about halo structure. Several theoretical works [38,39] have investigated the B-decay of 11Li into 9Li and a deuteron (d). They concluded that the fl-decay matrix elements are to a large extent determined by the halo part in 11Li. Experimental efforts in this direction reported in [40,41] and more recently in [42] provide evidence that the fi—decay takes place essentially in the halo of 11Li, and that it proceeds mainly to the 9Li + d continuum, opening up a new means to study the halo phenomenon in 11Li. The early reaction experiments were extended in later years, see for example reviews in [43—45]. They include transfer, stripping and break-up reaction studies providing differ- 6 ...=.Il git ential. mill att‘nttipiittii separation f for 11Li. 1111' 311i] i'iH'tI'lt' experiment s 111 5])lit‘l 1135 fliil lir‘t'l flt’lllft'ilih' ('I it valence iimir \‘i‘tlmt'ti part It.) a Strung [1.3th Tit.) quest :31 3;] Experimi bacnw st, (1, “J the (“built b9 turnip] 4“. nuclei as mh Strut-um. m“ [’liliis Work. exotic- IIII('1i -i ential, rather than integrated cross-sections. Reaction and decay experiments have been accompanied by precise measurements of static properties: measurements of two-neutron separation energy by methods of radio-frequency spectrometry [14] and Penning trap [46] for 11Li, nuclear charge radius determined by laser spectroscopy for 6He [47] and 11Li [48], and electric quadrupole and magnetic moments of 11Li from nuclear magnetic resonance experiments [49]. In spite of all the experimental efforts, the detailed structure of the two—neutron halo has not been deciphered yet. The consensus seems to be that, in 6He, the two maverick neutrons coexist anywhere between two extreme configurations [13]: a di—neutron with valence neutrons closely spatially correlated, and a cigar configuration in which the two valence particles are on opposite sides of the core. In 11Li, the situation is less clear due to a strong competition between s— and p—waves in the halo part of the wavefunction [13,50]. The question of clustering in 6He and 11Li is the subject of ongoing experimental quest [51—53]. Experimental data concerning two-neutron halos collected over the last decades has become so detailed that theoretical models must be more than merely qualitative to rise to the challenge. Even simple properties, such as the size of the nucleus, turn out to be model dependent and are not real experimental observables [54]. The study of halo nuclei as unstable species via reaction experiments involves tightly intertwined aspects of structure and reaction physics. Details of the reaction component are beyond the scope of this work. Nevertheless, reviews of reaction models used to probe the structure of light exotic nuclei can be found in [55,56]. 1.4 Overview of 6He and 11Li: structure theory Traditionally, theoretical considerations of structure and reactions of halo nuclei have been dominated by few-body models. Few-body structure models of two-neutron halo nuclei have built their success around the fact that, when viewed at a distance, the halo particles are decoupled from the core. Under such an approximation, the core’s degrees rival . .). [figure 1" nirit’li'is. a II‘ Offrofifiltllll (‘ part. The I: tt‘vfi'i‘illiir 1W mirn'm‘rvtiii' The mrl‘ St'llf"!llitll(' in Sophistimtwl his. mule-i the three-1): If deer apprt my method (In t! the (‘1Il.\tt'r-( ,; reported in '1 rewarded t hi" With lill'fi many~1ior1v p STUII‘TUI'P 111‘“ I i nit-tn t0 empl 1' 2 ’ l1.l(IlOIl Mann 1-, . TM: mode] ”'1 '11‘11 r -- ... .norli +1 :C‘fi Infidel wit it all O O :2 O O microscopic microscopic cluster three-body model model model Figure 1.2: Schematic representation of structure models of 6He. In microscopic cluster models, a microscopically described 4He core is formed explicitly. of freedom can be reduced, and the wavefunction factorizes into the core and the valence part. The many-body problem then reduces to a three-body one—core + n + n———held together by effective core-n and n-n interactions. For 6He, the transition from a fully microscopic to a few-body picture is schematically depicted in Figure 1.2. The early di-neutron models of 6He and 11Li, such as [31], turned out to be too schematic to quantitatively describe experimental data and were soon followed by more sophisticated three-body approaches. In the first generation, the three-body models of these nuclei treated the core as a completely inert object. Several methods of tackling the three-body problem were applied, mostly to 6He and 11Li. They include the Fad- deev approach [13,57,58], the hyper-spherical harmonics method [13,59], the variational method on a harmonic oscillator basis [60], the two-body Green’s function [61], and the cluster-orbital shell model [62,63]. Some calculations within a pairing model were reported in [64]. In all their generosity, the three-body models of the next generation rewarded the core with some degrees of freedom, namely with rotational modes [65]. With increasing computational power in recent years and new techniques to solve many-body problems, ab-initio microscopic competitors have emerged in the field of structure models of light exotic nuclei. The microscopic nature of these models allows them to employ realistic nucleon-nucleon and three-nucleon interactions. The Green’s function Monte Carlo model has been successfully applied to light nuclei up to 12C [66,67]. The model reproduced the three-body binding energy and radius of 6He. The no—core shell model [68] is another sophisticated approach, which as its name suggests, is a shell model with all particles active in harmonic oscillator shells; i.e. there is no inert core Evil Dim hit“, in statii and “Ll 7 dt’lldlzllli‘ 1" The) built 15 Their Hl’i’ii‘ .\11iit‘;t(’ (‘iifil dynamics mt Stilllt’W119' models. in \i' maitcls. This internal strut tun-s can aim Stut‘hdstit' m: helium [76: ill strength. the i and ”Li. (m. .111 these 111m] miitii‘in, Our nt'en'ii “Eddy f1i.\])i-r: Ullf‘it‘iins ll] flit Wan fields. p,“ appl‘lpriate. a. .11 s’irtm'er. t 1 1i“ 1; mean fielil (If 11 B, . irrommn nut-1 1:.»- Mill field cairn (trifle Vumtion ”11' it I a good (luv- Tl like in standard shell model calculations. The model has been applied to both 6He [69] and 11Li [70]. The fermionic molecular dynamics and the antisymmetrized molecular dynamics represent conceptually similar approaches to the problem of light nuclei [71]. They both use superpositions of Gaussian wave packets for single—particle wavefunctions. Their application to helium isotopes can be found in [72,73]. As in Green’s function Monte Carlo, the structure of 11Li has not yet been successfully described by molecular dynamics models. Somewhere between few-body and truly microscopic models are microscopic cluster models, in which some degrees of freedom are frozen to reduce the computational de- mands. This is achieved through the formation of microscopic clusters with a simplified internal structure within the nucleus being modeled. To a certain extent, cluster struc— tures can also be recognized in some of the micrOSCOpic models mentioned above. The stochastic variational model [74] and its multi-cluster version [75] has been applied to helium [76] and lithium [77] isotopes. With simpler phenomenological forces of adjusted strength, the model has been able to reproduce basic (three-body—like) properties of 6He and 11Li. Other examples of microscopic cluster models applied to 6He include [78—80]. All these models rely on Gaussians of one sort or another to describe the inter-cluster motion. Our overview of structure models would not be complete without mean field theories. Widely dispersed halo particles barely feel the short-range nuclear forces exerted by nucleons in the core. As a consequence, valence and core particles experience different mean fields. For Borromean systems in particular, the term mean field is probably not appropriate, as the correlations between halo particles are crucial for the overall binding. Moreover, the last neutron in the core + n + 11 system can not be bound in the localized mean field of the core + n subsystem since such a bound subsystem does not exist in Borromean nuclei. The importance of unusually small neutron separation energies for mean field calculations was recognized early on [81]. In later shell model calculations, configuration mixing and adjustments to residual interactions have become unavoidable for a good description of exotic nuclei towards the drip-lines [82,83]. In general, mean 9 pail field Alli—m W 1.5 m. The uhviulh exact ”Hill“ the relatin- ll ptupttlt. tiiv'j 1913111305. ()3; 11:11 temple ‘in ing t-nizipiimt: rtsults fur in. flit.iili.'1>' will \I. We are 11‘ mp9 with tit-i mentioned str {intents trmlir T110 ("rm-j“ dUuhle-(«lgpd q litittt‘C‘t‘n (1 m. ,1 Cation Of 1]“, II “13351 that (*u 1”“ 0n the h ()I 1“ I110 mlffilf‘] r .18. & ram . . attic halo It is al a. p, . Ilin tag from the Ra‘s O f )y .u ( A field approaches have found it rather challenging to obtain a reasonable description of halo effects in light nuclei. 1.5 Motivation for present work The obvious advantage of few-body structure models of two-neutron halo nuclei is the exact treatment of halo dynamics. These models provide clear, intuitive insight into the relative motion between the core and valence particles, and as long as implemented properly, they are well suited to capture the long-distance halo characteristics and cor- relations. On the other hand, micrOSCOpic models tackle the many body problem in its full complexity. Thanks to our advancing knowledge of nuclear interactions and increas- ing computational power, brute force ab-initio models now yield very accurate structure results for many light nuclei. It is reasonable to believe that, sooner or later, ab—initio models will succeed in producing an accurate description of halo nuclei. We are now in the position to ask why we need yet another structure model to cope with two-neutron halo nuclei. The answer is buried in drawbacks of the above- mentioned structure models and the lack of connection of some of them to reaction theories traditionally formulated in a few-body framework. The crucial assumption of few-body models, a macroscopic core, turns out to be a double-edged sword: on one hand, it allows us to focus on the most important correlations between core and halo nucleons, on the other hand, it is undoubtedly a (crude) simplifi- cation of the many-body problem. To argue in favor of inert cores, some authors indeed suggest that core polarization in halo nuclei is suppressed compared to normal nuclei [84]; but on the other side, there are works that admit the possibility of less inert cores inside halo nuclei [48,76]. Despite the occasional strong claims by few-body practitioners [85], realistic halo nuclei are unfortunately not ideal halo systems; the simple halo picture is always obscured by small idiosyncrasies, and one has always to check that the core is really unperturbed to justify the simplified inert-core few-body approach [86]. Stem- ming from the simplified picture of the core, probably the two most severe drawbacks 10 .ail Qt U A "‘2' :n": ‘ of few-iv "1‘ inwramiuti‘ amid-nun“? Prm'iiii" (iiifi halt) 1”m ”1' rpprntlut‘i‘ *4 of the win '1" fort-cs. Fur? ii intuitions 1’“ missing in ii; Lard in hit M Sumo of I gr-ttipit‘ 1(‘1‘:I>l The Hlli'fll.“ use phenom is wrijmg at The first out unrlvrstanriii picture. our the full llllt' E‘v'f’ll lilting] Shin Wit [“1 PTOYirliIig M Complt’ldl it ,1 adf-‘qliar-y l’)‘. PITJTIL I} I: 2‘ " mite (alt-til b. ‘5‘" ' ' ‘3- Clit‘.‘ is I. ‘— AIIJXI‘T'.\.‘ of few-body models are the lack of exact antisymmetrization and the usage of effective interactions [87]. Several Pauli blocking techniques have been developed to account for antisymmetrization in few—body models, however, when compared side by side, they may provide different results [88]. Effective interactions, especially those between the core and halo particles, are not necessarily known. Normally, the core-n potentials are adjusted to reproduce some set of experimental core-n findings and the three-body binding energy of the whole nucleus, or attempts are made to derive them from the underlying nuclear forces. Furthermore, there are indications that for reaction calculations three—body wave— functions perhaps require additional renormalization to account for microscopic effects missing in the inert-core approximation [89]. Nevertheless, few—body models are presently used in most reaction calculations involving halo nuclei. Some of the above-mentioned drawbacks of few-body models are eliminated in micro- scopic (cluster) models with halo particles made indistinguishable from those in the core. The microscopic treatment allows one to antisymmetrize wavefunctions properly and use phenomenological or realistic nucleon-nucleon (and three-nucleon) forces. So, what is wrong with microscopic (cluster) models? Well, one could object to several things. The first one is the missing connection to reaction theories, a link so important for the understanding of halo species. To feed reaction calculations formulated in a few-body picture, one would have to extract the necessary information about halo particles from the full microsc0pic wavefunction, a task that is by no means trivial computationally. Even though recently we have witnessed some progress in this direction for two-body-like (but not halo) projectiles [90,91], most microscopic structure theories are still far from providing such few-body-like information relevant for three-body-like halo nuclei. This computational obstacle is accompanied by a more fundamental physics question of the adequacy of microscopic models in the asymptotic regions. Horn the previous short review of structure theories it has become obvious that to make calculations feasible microscopic (cluster) models exploit computationally tractable bases. Chief among them are the Gaussians and harmonic oscillators. One must remem- ber, however, that at large distances, where the halo nucleons are almost liberated from 11 — 1'11 Qt 1119 (1UP. 1 capture th as argued itrds hint? tit'itis their: retreat 11211 are in gritt- 1111('T'I>('Ii]lil. of ('UIlW‘TEII" antw this (' Ravehinrth, 1391,5011 it; crt;rsttipir q; “1‘11 to mix l‘v‘t'i'I-nplnr‘ Ill ITllST‘dIIi‘i" Hui CUH‘II31 (if it Wat't‘flillifiitm halt) lllli'll’l tit 1121108). In 111(‘11. . Pff)tll.l(‘l (if d n 0 ~ . . txhonmrit-t‘ r I it " ' tsptttt of lilt‘ ] m;- ‘ it): flllf‘lf‘UIl‘ 3M1? . 1 I ' 1 1’15; ()f “(1 the .. mdW’fllIirt [I the core, the wavefunction falls off exponentially. In principle, it should be possible to capture the slower exponential decay by using a large Gaussian or oscillator basis, but as argued in [5], quality precedes quantity in the halo world; that is the actual shape of basis functions matters more than the size of the basis. In other words, the basis func- tions themselves ought to possess the correct long-distance functional form to produce correct halo asymptotics. For this reason, the authors of [5] concluded that Gaussians are in general not at all suited as a computational basis for halo nuclei. Moreover, most microscopic calculations are variational with the binding energy used to assess the rate of convergence. In general, the convergence of the total binding energy does not guar- antee the convergence of other observables and definitely not the convergence of the wavefunction in asymptotic regions. Based on the arguments presented, one can conclude that both few-body and mi- croscopic structure models have their appealing aspects as well as their drawbacks. We wish to mix the best of the two approaches to create a microscopic structure model of two-neutron halo nuclei that would describe simultaneously short- and especially long- distance regions and allow us to link .the structure and reactions of these nuclei. The concept of a microscopic cluster model with a carefully chosen functional form for the wavefunction seems to be ideal to meet our goals. Hereafter, the model of two-neutron halo nuclei developed in the present work shall be referred to as MiCH (microscopic core halos). In MiCH, a two—neutron halo nucleus will be described by a properly antisymmetrized product of a microscopic core and the valence part consisting of two individual neutrons, or schematically II! = Acore‘val(core x valence). We shall use terms “core” and “valence” in spite of the presence of the core-valence antisymmetrizer Acme-"al which, in principle, makes nucleons from the two parts of the wavefunction indistinguishable. A more precise meaning of “core” and “valence” will be provided as we go along. At large distances, the wavefunction naturally decouples into the three-body—like form \I! ——+ core x n x n, whereas at short distances it is equivalent to a fully antisymmetrized, many-body treat- ment. To bind the nucleus, effective nucleon-nucleon interactions shall be employed. The 12 P-al lllf‘lfify (1i’\‘r‘ nuclei. The ttt‘t.“i-Iii;*ltt rt iii in one step prams is iii 112110 pit ’11“ 1.6 0‘ 1n the prwr 111-3616 Hi the haptvr '2 it” PHIPIIIS of it} 10ml for 111i.“ 118 nt‘utrtitis 6H9 and 111.1 mtiflt‘l that lit Chapter 4 by 1119 COII‘iputgn tron of vari B ill it 2151c strtlt't lit a .. re (‘Olllpitl‘t‘ll .11th.\$1(ill. tht S’fl - ' t .irture input outlook in f '1 t.- theory developed in this work is designed to cope with bound states of two-neutron halo nuclei. The link between structure and reactions will be established for simultaneous two-neutron transfer. In this reaction channel, the two valence neutrons are transferred in one step from a halo projectile to a target nucleus. The transition probability of this process is directly proportional to the overlap integral between the original two-neutron halo projectile and its own core. 1.6 Outline In the present work, we elaborate on all aspects of MiCH. First, the two major building blocks of the wavefunctions—the valence part and the core—are discussed separately. Chapter 2 focuses on the valence part. A particular three-body model is described, el- ements of which are later incorporated into MiCH. To find the appropriate functional form for the valence part, the three—body dynamics between the core and the two ex- tra neutrons is studied in interaction-free regions. Chapter 2 also contains results for 6H6 and 11Li studied within a three-body approach. Chapter 3 presents a microscopic model that meets requirements imposed on the core. MiCH is then finally assembled in Chapter 4 by putting the core and the valence part together. That chapter also includes the computational background needed for evaluation of matrix elements and optimiza- tion of variational parameters. Chapter 5 contains results for 6He studied within MiCH. Basic structural features of 6He are elaborated on, and the results obtained within MiCH are compared to those from other models and to experimental data. As part of the discussion, the two-neutron transfer reaction p(6He,4He)t is studied using microscopic structure input for 6He modelled in MiCH. The work finishes with the conclusions and outlook in Chapter 6. 13 ...;I Chap1 Valen' Con: retry t«. l s bound and t 1 serrahlt‘s. Atty at‘t‘tjitiiit the: 111: (iii? Sllltjt‘l'l in tiUC‘li.1.tlt iIltt‘I'm to pin tit with it: added Bum )Iiii any other my“ with the few-1n the Wat'efunt't it CUT" 211111 a lllhw neutrons rt-lut it. 111 1110 r-m‘rt Established thri- model will 1w r PM thrtre ht it 1 V mode . ingrmlit 'ti Three-1 ‘ .. ind)“ Intuit. Chapter 2 Valence part Contrary to standard nuclei, valence particles in two-neutron halo nuclei are weakly bound and the tail of the wavefunction offers large contributions to most physical ob— servables. Any structure model aimed at the description of halo species should take into account the fact that the loosely bound neutrons swim in distant, low-energy regions and are subject to an interaction which is closer to the free rather than in—medium nucleon- nucleon interaction. Thus, a proper treatment of the asymptotic regions is vital if one is to pin down any observable sensitive to the spatial extent of the nucleus. Moreover, the added Borromean peculiarity of two-neutron halos implies pure three-body rather than any other asymptotics. Few-body models are especially well suited to cope successfully with the few-body dynamics and asymptotics of two-neutron halo nuclei. In Chapter 4, the wavefunction in MiCH will be cast as an antisymmetrized product of a microscopic core and a three-body-like valence part describing the relative motion of the two valence neutrons relative to the core. In the current chapter, we focus on the valence part. To do so, we outline a well established three—body model [65,92,93]. To avoid repetition, the mentioned three-body model will be referred to as “the three-body model”. First, we introduce coordinates and three-body basis sets used to attach the halo neutrons to the core in the three-body model, ingredients to be incorporated later into MiCH. Then, we outline details of the three-body model beyond what will be built into MiCH, such as interactions, the Pauli 14 ..-;I principle. anti will have no it than out in it] i: are inclnrlml it: plii‘iiishi‘ti lit r: in :11 1: an: reign flt‘t’l'lt‘fl I‘lr 1“ '1: 2.1 Coc The key inqrm spherical ft irnm Dh‘CSits. W115 1m rial tit in the a try tii'iurtnittit Ill 1111 hyper-radial 1n Laznt'rre hyper for (“lam y. i chapter. "\‘ali‘nt the Pmpcrt it *.\ ( , . . fUll informat it ill the meaning of T0 50!? ht iv: tare ‘9‘ n + n b TUlf . l 1] 11.“(1dr(‘ \‘)] wt: Spurious mm ”11*? at d “Puff”! N )1] . ‘ g fringe (. (if. principle, and the actual way of solving the three-body problem. In later chapters, we will have no use of these extra aspects of the three-body problem, but it is useful to lay them out before us to perform three-body calculations for 6He and 11Li, results of which are included in this chapter. The results of three-body calculations for 11Li were recently published by the author and collaborators [50]. For 6He, calculations originally published in [94] are repeated to reach results that were not included in that article but that are needed for comparison with results obtained within MiCH for this nucleus. 2.1 Coordinates and bases The key ingredient of the three-body model is the Schr6dinger equation in the hyper- spherical formalism. The hyper-spherical method, which had been used in other areas of physics, was brought into nuclear physics in [95] with the aim to develop a general nuclear reaction theory. The value added to three-body models in [65,92] was the introduction of deformation and rotational degrees of freedom to an otherwise inert core. The Sturmian hyper-radial basis exploited in [65, 92] was later in [93] replaced by a more suitable Laguerre hyper-radial basis [96]. For clarity, we should define terms “core” and “valence” more precisely. In the current chapter, “valence” will refer to all features of the three-body core + n + n system except the properties of the core, i.e. it will encompass spins of the two neutrons as well as the full information about the relative motion between the three bodies. Later, in Chapter 4, the meaning of these terms will be elaborated. To see how the three-body model is assembled, let us first analyze a three-body core + n + n bound problem in interaction-free asymptotic regions1 where, as argued in [5], one ought to employ a basis with appropriate exponentially decaying form. At this point, we are solely interested in relative motion between the three bodies. To eliminate the spurious motion of the total center of mass, only relative Jacobi coordinates between core and neutrons are used as shown in Figure 2.1. In principle, the two sets of Jacobi 1Long range Coulomb effects are absent due to charge neutrality of valence particles. 15 ..-:l (tit‘itt‘litlati's cast in any ill rttx‘irtlinatts l5 Partit‘lp'llkp 1) Width in 1m 1 a: \ Wllll lightly < 1! is WNW“ par1511li‘l'llll]litt it"‘r' ' .19“? 1]- ari‘ at) tall ‘dllll they ' momenta 1: and particle t-oort ll 11' IL"; , an advantage they allow the t rat '~ lidl equation. 1 .n . ~ ,ptrsphcrit‘al r I 31 ' iii-trig- coordinates—~Y and T—are completely equivalent and the three-body problem can be cast in any of them. The main advantage of Jacobi coordinates over other sets of relative coordinates is that the operator of kinetic energy decouples into two independent single- particle—like pieces with no cross term: it? 1 1 h? T=——- —A~ +—A,~,2 = 2m #1 1.1 #2 [A5 + A37] , (2.1) 7275 where m is the mass of a nucleon. Then, the interaction—free three-body Schréidinger equation becomes: T12 (5,37) = E3body¢ (13,37), (2-2) with E3body < 0 being the three-body binding energy. It is convenient to seek the solution of Eq. (2.2) in the form with angular and radial parts decoupled, schematically: 12(59') 2 HOT, 3»le (931223,,(931), (23) where Y; are spherical harmonics (for now, their projection quantum numbers are omit- ted) and they take care of the angular part of Eq. (2.2). We stress that the orbital momenta la; and ly are associated with Jacobi coordinates, rather than any sort of single- particle coordinates. Next, hyper-spherical coordinates from Figure 2.1 are involved. The main advantage of hyper-spherical coordinates is that, as it will soon become obvious, they allow the transformation of the original Eq. (2.2) into a one-dimensional, hyper- radial equation. The radial function H(:r,y) can be equally well written in terms of hyper-spherical coordinates p and 0, i.e. ’H($,y) = ’H(p,l9). Plugging (2.3) into (2.2) yields: a 1 — (p555) + $19] H (12.6) = EgbodyH (a9), (2.4) 16 / I Let us ("HIM I ohjett: are ilk? N'l_~ If. ’— RPIMIW‘ .14“ 1 tilt" la>t J.“ n t'vrztcr of rm: ‘ H l l H: I.) l \ Th“ VUEHHU‘ (' litre. Q, (., ,m] II. ‘\(’xt. r.:\(.‘ Wllll (limvnsin \ \ \‘ NUtp . 0 -\thdl Shh "I \ ( .‘ '21 RIM] and ”If“ ln'p. ‘r- Let us consider a three-body system core + II} + n2. In the laboratory frame, the three objects are at positions 77007.8, 7"}, and 7"}, . Then, there are two different—Y- and T- like—sets of Jacobi coordinates :75 = {in = 1, 2, 3}: Relative Jacobi coordinates 531 and f2 connect centers of masses of subgroups of objects; the last Jacobi coordinate 53 (not shown in the graphics) is equal to the position of the center of mass of the three-body system in the laboratory frame: Y T 51 = in] — 77core 531: F112 _ 71.721 _. _. (77111 + Acorefowre)/(Acore + 1) 52 = (F712 + Fn1)/2 — Fcore 1172 = n2 — 53 = 77CMS = (Acorefcore + F721 + Fn2)/A The volume element corresponding to the two relative Jacobi coordinates is: dV = dildfg = x? 323 dxldxgdflldflg. Here, (2,- comprises the standard polar and azimuthal spherical angles associated with 5,7. Next, rescaled relative Jacobi vectors are defined as: 5=VH15L .77=\/M2$2 with dimensionless reduced mass factors: Y T #1 = Acme/(Acme +1) #1 = 1/2 #2 = (Acore +1)/A #2 = 2Acore/A Note that spherical angles associated with :i" and 37 are the same as {21 and (22, i.e. (2x = (21 and fly = {22. Finally, the hyper-spherical coordinates, the hyper-radius p and the hyper-angle 0, are introduced as: x=psin0, y=pcos€. The volume element now becomes: dV = (p1,u2)_3/2 p5 sin2 6 cos2 6 (1de ML; de. Figure 2.1: Definition of Jacobi and hyper-spherical coordinates for a three-body system core + n1 + 112. Acme and A are the mass numbers of the core and of the whole system. 17 .-al gu with the gI‘H- Tlds upwmm 'llifllllt’lll it it.‘ spa-[mm ..f e‘ In 93 flu: v. For the purpn me rt'laIiumlA -1. as. _ I R hvre PHI - Jtll‘ lElllf‘S of the h‘ For - d myt,“ I)” (..- Sliilf’rlm] Wlllllll with the grand-angular operator A2: 2.5 —2—n 26 6_6 86 sin2 6 cos2 6 ( ) This Operator contains dimensionless magnitudes ($0,; + 1) and ly(ly + 1) of orbital momenta as traces of the orbital motion. The grand-angular operator has a complete spectrum of eigenfunctions enumerated by hyper-momentum K: A2